In the partitive model, we interpret 124 as dividing 12 into four equal groups. The result is the number in each group. Here, there are four columns, so a natural division is to group each column. The result is 3, the number in each column.

b.

In the quotative model, we interpret 124 as dividing 12 into groups containing fours. The result is the number of groups. Here, there are four in each row, so a natural division is to group each row. The result is 3, the number of groups required to make 12.

The correct answer is (2) and (3), which each illustrate the multiplication problem 2  6 (i.e., two groups of six). The first model, (1), illustrates the problem 3  4 (or 4  3), and the last model, (4), illustrates the problem 6  2 (i.e., six groups of two). Note that all four models represent the same solution, 12, which does not imply that they represent the same problem.

Using the area model, we start with one flat, nine longs, and five units, and we wish to make a rectangle with 13 rows.

This cannot be done without first exchanging one of the longs for 10 units.

After doing this, we can arrange a 13-by-15 rectangle, so the result of the division is 15.

Using long division, 13 goes into 19 once, with a remainder of 6. Carry down the 5 for 65. Thirteen goes into 65 five times evenly. The quotient is 15.

The area model relates to long division in that it matches the long division algorithm. In long division, you first subtracted 130, or 10  13, and then subtracted 65, or 5  13, for a total result of 15. Thus, 15  13 = 195. Similarly, the area model gave you the rectangle 13 by 15, which consists of two rectangles: 10  13 and 5  13.

In problems like this one, there is no way to "subtract" chips that are not there. Here, adding four zero pairs does not change the value of +2, but it gives us the four red chips we need in order to "subtract."

To use the colored-chip model for division, we need to use either partitive or quotative division to do the computations. When dividing a positive number by a negative number, we cannot use either of those models. We cannot "partition" a group into a negative number of parts, nor can we "count" the number of negatives within a positive number.